The b quark mass from lattice nonrelativistic QCD

Transcription

1 Alistair Hart a, Georg M. von Hippel, R. R. Horgan c, Andrew Lee c, Christopher J. Monahan c a SUPA, School of Physics and Astronomy, University of Edinurgh, Edinurgh EH9 3JZ, U.K. Institut für Kernphysik, Universität Mainz, Becherweg 45, 5599 Mainz, Germany c DAMTP, University of Camridge, Wilerforce Road, Camridge CB3 WA, U.K. C.Monahan@damtp.cam.ac.uk, a.hart@ed.ac.uk, hippel@kph.uni-mainz.de, R.R.Horgan@damtp.cam.ac.uk, A.Lee@damtp.cam.ac.uk We present the first two-loop calculation of the heavy quark energy shift in lattice nonrelativistic QCD (NRQCD). This calculation allow us to extract a preliminary prediction of m (m, n f = 5) = 4.25(12) GeV for the mass of the quark from lattice NRQCD simulations performed with a lattice of spacing a =.12fm. Our result is an improvement on a previous determination of the quark mass from unquenched lattice NRQCD simulations, which was limited y the use of one-loop expressions for the energy shift. Our value is in good agreement with recent results of m (m ) = 4.163(16) GeV from QCD sum rules and m (m, n f = 5) = 4.17(25) GeV from realistic lattice simulations using highly-improved staggered quarks. We employ a mixed strategy to simplify our calculation. Ghost, gluon and counterterm contriutions to the energy shift and mass renormalisation are extracted from quenched high-eta simulations whilst fermionic contriutions are calculated using automated lattice perturation theory. Our results demonstrate the effectiveness of such a strategy. The XXVIII International Symposium on Lattice Field Theory, Lattice21 June 14-19, 21 Villasimius, Italy Speaker. Current address: Cray Exascale Research Initiative, JCMB, King s Buildings, Edinurgh EH9 3JZ, U.K. c Copyright owned y the author(s) under the terms of the Creative Commons Attriution-NonCommercial-ShareAlike Licence.

2 1. Introduction The precise theoretical and experimental determination of quark masses is an important component of high-precision tests of the Standard Model of particle physics. One current focus for tests of the Standard Model is the unitarity of the Caio-Koayashi-Maskawa (CKM) matrix, which descries flavour-changing quark transitions. Quark masses serve as an input into the tests of CKM matrix unitarity; the mass of the quark is used in the extraction of the CKM matrix element V u from inclusive semileptonic decays of B mesons [1]. Recent high-precision calculations of the quark mass using realistic lattice QCD simulations [2] and perturative QCD comined with experimental results [3] are in good agreement, otaining values of m (m, n f = 5) = 4.17(25) GeV 1 and m (m ) = 4.163(16) GeV respectively. For the first time, the lattice result was otained using the same action, the highly improved staggered quark (HISQ) action, for oth valence and sea quarks. HISQ is a highly corrected version of the standard staggered quark action that retains a chiral symmetry on the lattice [4]. Most current lattice studies of quarks use an effective field theory, such as nonrelativistic QCD (NRQCD), for the valence heavy quark. Simulating oth valence and sea quarks with the same action allows a much greater precision, ut is only now ecoming possile with the advent of finer lattices and highly improved actions. However, even on the very finest lattices with HISQ heavy quarks an extrapolation to the heavy quark mass is still required [2]. Our calculation improves on a previous determination of m (m ) = 4.4(3) GeV from unquenched lattice QCD simulations using NRQCD valence quarks [5]. The dominant error in that calculation arose from the use of one-loop perturation theory in the matching etween lattice quantities and the continuum result. By introducing a mixed strategy incorporating high-eta quenched simulations and automated lattice perturation theory, we perform the first ever such twoloop calculation in NRQCD. This serves a two-fold purpose. Firstly our calculation demonstrates the effectiveness of employing an efficient mixed perturation theory/high-eta simulation method for higher order perturative quantities. Secondly our result allows us to otain a more precise prediction for the quark mass from lattice NRQCD simulations. 1.1 Heavy quarks on the lattice Currently availale lattices are too coarse to directly simulate quarks, ecause the Compton wavelength of the quark is smaller than the lattice spacing. One common approach to solving this prolem is to introduce a nonrelativistic effective action, NRQCD, for which the discretization errors are under control and which can e systematically improved y including extra operators. NRQCD is constructed y integrating out dynamics at the scale of the heavy quark mass and then using the Foldy-Wouthuysen-Tani transformation to write the action as an expansion in the inverse heavy quark mass [6]. We use an NRQCD action correct to O(1/m 2,v 4 ), where v is the relative internal velocity of the ound-state heavy quarks. A detailed derivation of the action we use is given in [7]. The lattice NRQCD action can e written S nrqcd = ψ + (x,τ)[ψ(x,τ) K(τ)ψ(x,τ 1)], (1.1) x,τ 1 Note that Reference [2] quotes only the result at a scale equal to 1 GeV and the value of m (m, n f = 5) was otained using continuum perturation theory for the running MS mass. 2

3 with ( K(τ) = 1 δh 2 )( 1 H ) n ( U 4 1 H ) n ( 1 δh 2n 2n 2 ). (1.2) Here the leading nonrelativistic kinetic energy is H = (2) /2M. The correction term δh contains higher order terms in the 1/M expansion: the improved chromoelectric and chromomagnetic interactions, the leading relativistic kinetic energy correction and discretization error corrections. The integer n is introduced as a staility parameter. 2. Calculating the quark mass Quark confinement ensures that quark masses are not physically measurale quantities, so the notion of quark mass is a theoretical construction. A wide range of quark mass definitions exist, often tailored to exploit the physics of each particular process. One common choice of quark mass is the pole mass, defined as the pole in the renormalized heavy quark propagator. However, the pole mass is a purely perturative concept and suffers from infrared renormalon amiguities [8 1]. To avoid these amiguities, experimental results are usually quoted in the modified Minimal Sutraction (MS) scheme, which is renormalon amiguity free. Lattice calculations use the renormalon-free are lattice mass. These different quark mass definitions must e matched to enale meaningful comparison. We match are lattice quantities to those in the MS scheme using the pole mass as an intermediate step. Any renormalon amiguities cancel in the full matching procedure etween the lattice quantities and the MS mass. We extract the MS mass from lattice simulation data in a two-stage process. We first relate lattice quantities to the pole mass and then match the pole mass to the MS mass evaluated at a scale equal to the quark mass. 2.1 Extracting the pole mass We determine the pole mass using two independent methods. The first method relates the pole mass, M pole, to the experimental mass, M expt = 9.463(26) GeV [11], using the heavy quark energy shift, E : 2M pole = M expt (E sim () 2E ). (2.1) Here E sim () is the energy of the meson at zero momentum, extracted from lattice NRQCD simulations. The quantity (E sim () 2E ) corresponds to the inding energy of the meson in NRQCD. We use a value of E sim =.515(3) GeV, otained from a lattice NRQCD simulation run y the HPQCD collaoration on a coarse MILC ensemle, with lattice spacing a = 1.647(3) GeV 1 [12]. For further details of the configuration ensemle see [13, 14]. The second method directly matches the pole mass to the are lattice mass in physical units, M latt (a), via the heavy quark mass renormalisation, Zlatt M, M pole = Z latt M (µa,m latt We employ a mixed strategy to calculate E and Z latt M (a))mlatt (a). (2.2) perturatively. The fermionic contriutions to E, shown on the left-hand side of Figure 1, are calculated using two-loop automated lattice perturation theory. All other contriutions, shown on the right-hand side of Figure 1, are extracted from high-eta quenched simulations. 3

4 Figure 1: Contriutions to E and ZM latt. The four fermionic contriutions calculated using automated lattice perturation theory are shown on the left. The diagrams on the right are extracted from high-β simulation. Blue lines are heavy quarks, green are gluons and red are sea quarks. Large rown los represent the 5 gluon self energy diagrams and crosses are counterterms. Feynman diagrams reproduced from [15]. Results were otained using the NRQCD action of Equation 1.1 for the heavy valence quark, HISQ light quarks and the Lüscher-Weisz action for the gluons [16, 17]. We used a heavy quark mass in lattice units of Ma = 2.8, with a staility parameter of n = Automated lattice perturation theory Feynman rules for the NRQCD and HISQ actions are too complicated to e vialy derived y hand and the resulting Feynman integrals can only e evaluated numerically. We therefore use automated lattice perturation theory, employing HiPPy to derive the Feynman rules and HPsrc to evaluate the four diagrams [18, 19]. To control the highly-peaked IR ehaviour of the Feynman integrands, we introduce a gluon mass. Although in general a non-zero gluon mass cannot e used in calculations eyond one-loop, this issue concerns only diagrams containing ghost-gluon vertices. In our calculation, these diagrams are handled y the high-eta simulation, allowing us to use a gluon mass for the fermionic contriutions. The light quark diagrams in Figure 1 were calculated using five different light quark masses and extrapolated to zero light quark mass. We verified that the appropriate Ward identity for the 1-loop gluon self-energy was satisfied High-eta simulations We perform quenched simulations on L 3 T lattices with temporal extent T = 3L, for L = 3 to L = 1 and twisted oundary conditions to reduce finite size effects and tunnelling etween QCD vacua [17]. We generate ensemles of configurations for 17 values of β from β = 9 to β = 12. Since the Green function is not gauge-invariant, we fix to Coulom gauge using a conjugate gradient method. To extract the energy shift and mass renormalisation, we use an exponential fit to the heavy quark Green function parametrized as ( ] ) G(p,t) = Z ψ exp [E + p2 2ZM lattm +... t, (2.3) 4

5 where the ellipsis stands for higher order terms that are included in the fits. All operators in the NRQCD action are expressed in terms of gauge-covariant Wilson paths generated using PYTHON, which greatly enales flexiility and reduces programming errors. The heavy quark source is classified in the flavour-smell asis appropriate to the twisted oundary conditions. We implement the oundary conditions using a gauge-twist mask whenever a path in an operator crosses any spatial oundary. By applying an extra U(1) phase in the mask, we can assign an aritrarily small momentum to the source, enaling oth E and ZM latt to e relialy extracted as a function (β, L). We convert β to α V and perform a joint fit to extract the 1- and 2-loop coefficients in the L limit. Simulations were run including tadpole improvement, which significantly reduces the magnitude of oth 1- and 2-loop coefficients. Results for ZM latt are good ut this work is still in progress and we concentrate here on those for E. For even L, in Tale 1 we compare the tadpoleimproved 1-loop coefficient from an unconstrained fit to the simulation data for E with the exact result from automated perturation theory. To extract the 2-loop coefficient we constrained the 1- loop coefficient to e the exact value, ut Tale 1 shows that the simulation relialy reproduces the 1-loop results. The numer of independent configurations for each (β, L) was aout 3, which we can easily increase y 1-fold or more, allowing for much more accurate results at the next stage. L E sim..5295(16).5988(16).6369(12).656(11).738(63) E th (3) Tale 1: Comparison of an unconstrained fit from simulation for the perturative 1-loop coefficient with the automated perturative calculation. There is no error on the theory calculation as it was done y mode summation. The error on the theory extrapolation to L = is estimated from a fit. 2.2 Matching the pole mass to the MS mass Although the pole mass is plagued y renormalon amiguities, these amiguities cancel when lattice quantities are related to the MS mass. This renormalon cancellation is evident in the direct matching of the are lattice mass to the MS mass, as oth M MS and M latt M MS (µ) = Z latt M (µa,mlatt (a))z 1 cont (µ,m pole)m latt (a), (2.4), relates the pole mass to the MS mass and has een determined to O(αs 3) [2]. To see that renormalon amiguities also cancel in when determining the pole mass from the energy shift, we equate Equations 2.1 and 2.2 and rearrange them to otain are renormalon-free. The continuum matching parameter, Z cont M 2(ZM latt M latt (a) E ) = M expt E sim (). (2.5) The two quantities on the right hand side of the equation are renormalon amiguity free: M expt is a physical quantity and E sim () is determined nonperturatively from lattice simulations. Any renormalon amiguities in the two power series, ZM latt and E, on the left-hand side of the equation must therefore cancel. 5

6 3. Results For the fermionic and quenched contriutions to the two-loop heavy quark energy shift we find E =.7348(3)α V (q /a)+(1.37(6).23(1)n f )α 2 V(q /a)+o ( α 3 V). (3.1) We express our result in the V -scheme at a scale q /a = 3.33, a value determined using the BLM procedure in [21]. The uncertainties quoted for the one-loop coefficient and the quenched contriution to the two-loop coefficient arise from the multi-polynomial fit. For the fermionic contriution to the two-loop coefficient, the quoted uncertainty is the statistical error in the numerical evaluation of the Feynman diagrams. We estimated the coefficient of the O ( α 3 s ) term from the quenched simulation fits as 1.(5). Inserting this result for the heavy quark energy shift into Equation 2.1 leads to our first preliminary determination of the quark mass: M MS ( M MS) = 4.25(12) GeV. (3.2) The error is an estimate of O(α 3 s ) contriutions, which dominate the uncertainty in our result. Uncertainties arising from systematic and statistical errors in the lattice results, E sim () and E, are 1%. We are unale to estimate the systematic error due to O(a 2 ) artifacts as we have not yet finished the calculation for smaller values of a; this work is in progress and entails working with different values of Ma in NRQCD. It should e noted that we used a value of E sim () that was generated from lattice NRQCD simulations using the action of Equation 1.1, ut with n = 4. From 1-loop calculations we estimate the errors associated with this mismatch to e much smaller than the dominant O(α 3 s ) error. However, this discrepancy will e corrected in future work. 4. Conclusion We have calculated the two-loop heavy quark energy shift in highly-improved NRQCD using a mixed approach comining quenched high-eta simulations with lattice perturation theory. This is the first determination of any heavy quark parameter eyond first-order perturation theory in NRQCD, and demonstrates that we are ale to extract a more precise prediction of the quark mass from lattice NRQCD simulations than has een previously achieved. Work is currently underway to complete our calculation of the mass renormalisation, ZM latt, and to extend our results to incorporate different heavy quark masses to extrapolate to a =. We also plan to increase the size of the ensemles used in the high-eta analysis y a significant factor. We expect these developments will improve further the precision of our result for the quark mass. Acknowledgements We would like to thank Christine Davies and Iain Kendall for providing HPQCD simulation data. We thank the DEISA Consortium ( funded through the EUFP7 project RI , for support within the DEISA Extreme Computing Initiative. This work has made use of the resources provided y the Camridge High Performance Computing service supported in part y the Science and Technology Facilities Council under grant ST/H8861/1. 6

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